## Gene interactions: a geometric approach

### Alex Gavryushkin

Joint work with:
• Kristina Crona, American U., Washington, DC, USA
• Bernd Sturmfels, U. of California, Berkeley, USA
• Ewa Szczurek, U. of Warsaw, Poland
• Niko Beerenwinkel, ETH Zurich, Basel, Switzerland

October 19, 2016

Don't worry.

### Epistasis

$$w_{11} + w_{00} = w_{01} + w_{10}$$

#### Deviation from the additive expectation of allelic effects:

$u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$

### Understanding three-way interactions

#### Marginal epistasis?

$u_{\color{blue}{0}11} = w_{\color{blue}{0}00} + w_{\color{blue}{1}00} + w_{\color{blue}{0}11} + w_{\color{blue}{1}11} − (w_{\color{blue}{0}01} + w_{\color{blue}{1}01}) − (w_{\color{blue}{0}10} + w_{\color{blue}{1}10})$

#### Total three-way interaction?

$u_{111} = w_{000} + w_{011} + w_{101} + w_{110} - (w_{001} + w_{010} + w_{100} + w_{111})$

#### Conditional epistasis?

$e = w_{\color{blue}{0}00} − w_{\color{blue}{0}01} − w_{\color{blue}{0}10} + w_{\color{blue}{0}11}$

### (Algebraic) Geometry sorts out the mess!

$e = u_{011} + u_{111}$

In general, the four interaction coordinates $$u_{011}, u_{101}, u_{110}, u_{111}$$ allow to describe all possible kinds of interaction!

There are 20 types of interaction and they are known as circuits to Algebraic Geometry 111 students

Yep, we've got the list!

\scriptsize \begin{align*} a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\ b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\ c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\ d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\ e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\ f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\ g&=w_{000}-w_{011}-w_{100}+w_{111} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\ h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\ i&=w_{000}-w_{010}-w_{101}+w_{111}\\ j&=w_{001}-w_{011}-w_{100}+w_{110}\\ k&=w_{000}-w_{001}-w_{110}+w_{111}\\ l&=w_{010}-w_{011}-w_{100}+w_{101}\\ \end{align*}
\scriptsize \begin{align*} a&= w_{000}-w_{010}-w_{100}+w_{110} & m&=w_{001}+w_{010}+w_{100}-w_{111}-2w_{000}\\ b&=w_{001}-w_{011}-w_{101}+w_{111} & n&=w_{011}+w_{101}+w_{110}-w_{000}-2w_{111}\\ c&=w_{000}-w_{001}-w_{100}+w_{101} & o&=w_{010}+w_{100}+w_{111}-w_{001}-2w_{110}\\ d&=w_{010}-w_{011}-w_{110}+w_{111} & p&=w_{000}+w_{011}+w_{101}-w_{110}-2w_{001}\\ e&=w_{000}-w_{001}-w_{010}+w_{011} & q&=w_{001}+w_{100}+ w_{111}-w_{010}-2w_{101}\\ f&=w_{100}-w_{101}-w_{110}+w_{111} & r&=w_{000}+w_{011}+ w_{110}-w_{101}-2w_{010}\\ \color{blue}{g}&\hskip{2pt}\color{blue}{=w_{000}-w_{011}-w_{100}+w_{111}} & s&=w_{000}+w_{101}+ w_{110}-w_{011}-2w_{100}\\ h&=w_{001}-w_{010}-w_{101}+w_{110} & t&=w_{001}+w_{010}+w_{111}-w_{100}-2w_{011}\\ i&=w_{000}-w_{010}-w_{101}+w_{111}\\ j&=w_{001}-w_{011}-w_{100}+w_{110}\\ k&=w_{000}-w_{001}-w_{110}+w_{111}\\ l&=w_{010}-w_{011}-w_{100}+w_{101}\\ \end{align*}

### This is known as Beerenwinkel-Pachter-Sturmfels approach,

which provides a complete picture of interactions!

### BUT

the approach is

• based on the availability of fitness measurements

• computationally feasible for up to four loci

### Problem 1: What if no (credible) fitness measurements are available?

Like in this malaria drug resistance data set:
Ogbunugafor et al. Malar. J. 2016

### Results at a glance

• We provide a complete characterization of fitness graphs that imply circuit interaction (think epistasis).
• Fitness graphs arise in competition-like experiments and include:
• Rank orders
• Mutation graphs

(Preprint(s) with Crona, Beerenwinkel, and others to appear)

### Characterization of epistatic rank orders

Theorem. Consider a biallelic $n$-locus system. The number of rank orders which imply $n$-way epistasis is: $\frac{(2^n)! \times 2}{2^{n-1}+1}$

Corollary. The fraction of rank orders that imply n-way epistasis among all rank orders is: $\frac{2}{2^{n-1}+1}$

### Applications

• HIV-1
• Antibiotic resistance

• Fungi
• Gut microbiome (with Will Ludington, UC Berkeley)
• Synthetic lethality
• Knockdown cell lines

Methodologically, this allows us to advise further measurements (experiments) for incomplete data sets. This exponentially reduce the number of potential experiments (see the HIV example).

#### Example: antibiotic resistance

Mira et al. PLOS ONE, 2015

### Results in more detail

Efficient methods for:
• Circuit interaction inference (including epistasis and three-way interaction) for total orders
• Complete analysis of partial orders (including mutation graphs) with "distance to interaction" inference
• Suggestions for possible completions in case of missing data and/or high uncertainty

Software (pre-release stage):
https://github.com/gavruskin/fitlands

### Problem 2: What if the number of genes (loci) is 20,822?

• 2^20820 of conditional epistases?
• 2^20820 measurements to estimate marginal epistasis?

Not in this life

### Concrete example: genome-wide RNAi perturbation screens

20,822 genes, 90,000 "trials" (siRNA's)

### Two ways out

1. Ask biologists about "the most important genes" that are main fitness drivers (like we did in the HIV study)
2. Add statistical assumptions, for example:
• Ignore higher-order terms in Taylor series approximations

• Structural hypotheses

(Ongoing work with Schmich, Szczurek, Beerenwinkel, et al.)